Question

A friend of yours is given the following problem:

You have 8 seemingly identical coins, among which you are told that one is a counterfeit. The counterfeit is known to be heavier than a true coin. You have access to a simple two-pan balance. What is the MINIMUM number of weighings you need to use in order to find the counterfeit coin?

Your friend's answer is the following:

I can identify the counterfeit coin by proceeding as follows:

Weighing 1:

Put 4 coins on one pan of the balance and the other 4 on the other pan

Whichever side is heavier contains the counterfeit coin: so now, you have discarded 4 coins from being potential counterfeit coins. You are left with 4 coins to look within.

Weighing 2

Put 2 coins on one pan of the balance and the other 2 on the other pan

Whichever side is heavier contains the counterfeit coin: so now, you have discarded 2 coins from being potential counterfeit coins. You are left with 2 coins to look within.

Weighing 3

Put 1 coin on each pan

Heavier one is the counterfeit coin

So your minimum number of weighings is 3

ANSWER THESE QUESTIONS, PLEASE: What can you tell about the reasoning of your friend? Is it correct or not? If it is correct, explain why. If it is incorrect, explain why. You have to make an unambiguous compelling justification of either way.

Answer 1

Yes , it is correct and it is one of the good and optimistic solutions to find minimum number of weighings. but it is not one and only optimistic solution to find minimum number of weighings . and also to find the solution these three weighings must be necessary because you can't find the counterfit coin in less than three weighings ,if you follow above procedure. here both maximum and minimum number of weighings is 3 . so above procedure is not much recommended but it is good solution not great.

There are few solutions like beginning with four coins .it means

step-1 : putting 2 coins on one pan and 2 of the remaining on the coins on the other pan .

step-2 : whichever side is heavier contains the counterfeit coin. discard the lighter one take heavier side two coins and put each coin of the two coins on each pan. so here you will find counterfeit coin in just 2 weighings and minimum number of weighings become 2.

step-3 : if both are balanced in step-1 then take remaining 4 coins  with 2 on one pan and other 2 on other pan.

step-4 :  Repeat step-2 here compulsory you will get counter feit coin because these are last coins . and here minimum number of weighings become 3.

I think my solution is better than the given solution because in my soluiton you may find counterfeit coin in 3 weighings or 3 weighings.

but in the given solution compulsory you will find counter feit coin in 3 weighings.

Answer 2

Yes , it is correct and it is one of the good and optimistic solutions to find minimum number of weighings. but it is not one and only optimistic solution to find minimum number of weighings . and also to find the solution these three weighings must be necessary because you can't find the counterfit coin in less than three weighings ,if you follow above procedure. here both maximum and minimum number of weighings is 3 . so above procedure is not much recommended but it is good solution not great.

There are few solutions like beginning with four coins .it means

step-1 : putting 2 coins on one pan and 2 of the remaining on the coins on the other pan .

step-2 : whichever side is heavier contains the counterfeit coin. discard the lighter one take heavier side two coins and put each coin of the two coins on each pan. so here you will find counterfeit coin in just 2 weighings and minimum number of weighings become 2.

step-3 : if both are balanced in step-1 then take remaining 4 coins  with 2 on one pan and other 2 on other pan.

step-4 :  Repeat step-2 here compulsory you will get counter feit coin because these are last coins . and here minimum number of weighings become 3.

I think my solution is better than the given solution because in my soluiton you may find counterfeit coin in 3 weighings or 3 weighings.

but in the given solution compulsory you will find counter feit coin in 3 weighings.